
where \(\Pi _\mathcal {X}(\cdot )\)  is the Euclidean projection onto \(\mathcal {X}\) , that ensures feasibility, and \(\eta \in \mathbb {R}_+\)  is called the learning rate. Note that the state \(\mathbf {x}_{t+1}\)  obtained according Eq. (REF ) is indeed a function of \(\lbrace (\mathbf {r}_{t}, \mathbf {x}_{t})\rbrace  \subset \lbrace (\mathbf {r}_{s}, \mathbf {x}_{s})\rbrace ^{t}_{s=1} \)  for every \(t \ge 1\) ; hence, OGD is indeed an online caching policy. Paschos et al. [1]}, show that OGD attains sub-linear regret when \(R=h=1\) ; more specifically, the following holds:
